Kolmogorovspectra

Kolmogorov spectra, also known as inertial range spectra, are a fundamental concept in turbulence theory, describing the statistical properties of turbulent flows at different scales. Proposed by Andrei Kolmogorov in 1941, these spectra relate the energy contained in turbulent eddies to the size of those eddies within a specific range of scales.

In a turbulent flow, energy is injected at large scales (e.g., from stirring or natural phenomena). This energy then cascades down through progressively smaller eddies. The inertial range is the region of scales where this energy transfer occurs most efficiently, without significant influence from either the energy injection scales or the smallest dissipative scales.

Kolmogorov's first and second similarity hypotheses are central to the concept. The first hypothesis states that the statistical properties of small-scale turbulence are independent of the large-scale structures and depend only on the mean rate of energy dissipation. The second hypothesis extends this to predict that the energy spectrum in the inertial range depends solely on the energy dissipation rate and the wavenumber (which is inversely related to scale).

This leads to the prediction that the energy spectrum, E(k), in the inertial range follows a power law: E(k) proportional to epsilon^(2/3) * k^(-5/3), where epsilon is the rate of energy dissipation and k is the wavenumber. This universal -5/3 power law has been widely observed in various turbulent phenomena, from atmospheric turbulence to ocean currents and pipe flows, making Kolmogorov spectra a cornerstone of understanding and modeling turbulent systems.